Perfect codes and minimal distance of a code have great importance in the study of theoryof codes. The perfect codes are classified generally and in particular for the Lee metric.However, there are very few perfect codes in the Lee metric. The Lee metric hasnice properties because of its definition over the ring of integers residue modulo q. It isconjectured that there are no perfect codes in this metric for q > 3, where q is a primenumber.The minimal distance comes into play when it comes to detection and correction oferror patterns in a code. A few bounds on the number of codewords and minimal distanceof a code are discussed. Some examples for the codes are constructed and their minimaldistance is calculated. The bounds are illustrated with the help of the results obtained.

A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.

Static analysis of database design and implementation is not a new idea. Many researchers have covered the topic in detail and defined a number of metrics that are well known within the research community. Unfortunately, unlike the use of metrics in code development, the use of these metrics has not been widely adopted within the development community. It seems that a disjunction exists between the research into database design metrics and the actual use of databases in industry. This paper describes new metrics that can be used in industry to ensure that a database's current implementation supports long term scalability, to support easily developed and maintainable code, or to guide developers towards functions or design elements that can be modified to improve scalability of their data systems. In addition, this paper describes the production of a tool designed to extract these metrics from SQL Server and includes feedback from professionals regarding the usefulness of the tool and the measures contained within its output. / text

As people go about their lives, they form a variety of social relationships, such as family, friends, colleagues, and acquaintances, and these relationships differ in their strength, indicating the level of trust among these people. The trend in these relationships is for people to trust those who they have met in real life more than unfamiliar people whom they have only met online. In online social network sites the objective is to make it possible for users to post information and share albums, diaries, videos, and experiences with a list of contacts who are real-world friends and/or like-minded online friends. However, with the growth of online social services, the need for identifying trustworthy people has become a primary focus in order to protect users’ vast amounts of information from being misused by unreliable users. In this thesis, we introduce the Capacity- first algorithm for identifying a local group of trusted people within a network. In order to achieve the outlined goals, the algorithm adapts the Advogato trust metric by incorporating weighted social relationships. The Capacity-first algorithm determines all possible reliable users within the network of a targeted user and prevents malicious users from accessing their personal network. In order to evaluate our algorithm, we conduct experiments to measure its performance against other well-known baseline algorithms. The experimental results show that our algorithm’s performance is better than existing alternatives in finding all possible trustworthy users and blocking unreliable ones from violating users’ privacy.

As people go about their lives, they form a variety of social relationships, such as family, friends, colleagues, and acquaintances, and these relationships differ in their strength, indicating the level of trust among these people. The trend in these relationships is for people to trust those who they have met in real life more than unfamiliar people whom they have only met online. In online social network sites the objective is to make it possible for users to post information and share albums, diaries, videos, and experiences with a list of contacts who are real-world friends and/or like-minded online friends. However, with the growth of online social services, the need for identifying trustworthy people has become a primary focus in order to protect users’ vast amounts of information from being misused by unreliable users. In this thesis, we introduce the Capacity- first algorithm for identifying a local group of trusted people within a network. In order to achieve the outlined goals, the algorithm adapts the Advogato trust metric by incorporating weighted social relationships. The Capacity-first algorithm determines all possible reliable users within the network of a targeted user and prevents malicious users from accessing their personal network. In order to evaluate our algorithm, we conduct experiments to measure its performance against other well-known baseline algorithms. The experimental results show that our algorithm’s performance is better than existing alternatives in finding all possible trustworthy users and blocking unreliable ones from violating users’ privacy.

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This thesis verified that metric spaces can be constructed using ultrametrics d and D, where d(x,y) = 0 if x = y and d(x,y) = (1/2) k if x not equal to y, such that x-y = 2k(a/b) for a,b relatively prime to 2, and where D(A,B)= max(d(al,bl); d(a2,b2)) for A = (al,a2) and B = (bl,b2).Assuming that a line is represented by some linear equation, a one-dimensional point was defined as an element of Q and a two-dimensional point as an element of Q x Q. There was an investigation of one-dimensional points with respect to the behavior of segments, midpoints, and distances as measured by d. The function D demonstrated the behavior of midpoints, medians, and triangles, as well as the congruence relation. The study necessitated the introduction of pseudomidpoints and pseudomedians, and an unorthodox definition of angle measurement.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, May 2008. / Title from title screen. Includes bibliographical references.

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Mathematics." Includes bibliographical references (p. 82-85).